How do you implicitly differentiate #-y= 4x^3y^2+2x^2y^3-2xy^4 #?
Implicit Differentiation is a special case of the chain rule. When you differentiate the y variable in a term or factor you differentiating with respect to x. Because of this you have include the factor of
We will have to use the product rule, power rule and chain rule.
Divide all the terms by -1 to remove the negative
Differentiate Implicitly Distribute the negative Gather the terms with the Factor out Divide to isolate Multiple numeric factors to simplify Below are a couple of tutorials include implicit differentiation
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To implicitly differentiate the given equation, follow these steps:
- Differentiate each term with respect to ( x ).
- Apply the product rule when differentiating terms that involve both ( x ) and ( y ).
Differentiating term by term:
For ( -y ), the derivative is ( -\frac{dy}{dx} ).
For ( 4x^3y^2 ), the derivative is ( 12x^2y^2 + 8x^3y\frac{dy}{dx} ) using the product rule.
For ( 2x^2y^3 ), the derivative is ( 4x^2y^3 + 6x^2y^2\frac{dy}{dx} ) using the product rule.
For ( -2xy^4 ), the derivative is ( -2y^4 - 8xy^3\frac{dy}{dx} ) using the product rule.
Combine all the terms:
[ -\frac{dy}{dx} = 12x^2y^2 + 8x^3y\frac{dy}{dx} + 4x^2y^3 + 6x^2y^2\frac{dy}{dx} - 2y^4 - 8xy^3\frac{dy}{dx} ]
Group terms involving ( \frac{dy}{dx} ) together:
[ -\frac{dy}{dx} - 8x^3y\frac{dy}{dx} + 6x^2y^2\frac{dy}{dx} - 8xy^3\frac{dy}{dx} = 12x^2y^2 + 4x^2y^3 - 2y^4 ]
Factor out ( \frac{dy}{dx} ):
[ \frac{dy}{dx}(-1 - 8x^3 + 6x^2y^2 - 8xy^3) = 12x^2y^2 + 4x^2y^3 - 2y^4 ]
Divide both sides by the expression in parentheses:
[ \frac{dy}{dx} = \frac{12x^2y^2 + 4x^2y^3 - 2y^4}{-1 - 8x^3 + 6x^2y^2 - 8xy^3} ]
This is the implicit derivative of the given equation.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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